# 5 1

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 5 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 5 1 at Knotilus! An interlaced pentagram, this is known variously as the "Cinquefoil Knot", after certain herbs and shrubs of the rose family which have 5-lobed leaves and 5-petaled flowers (see e.g. [4]), as the "Pentafoil Knot" (visit Bert Jagers' pentafoil page), as the "Double Overhand Knot", as 5_1, or finally as the torus knot T(5,2).

 A kolam of a 2x3 dot array The VISA Interlink Logo [1] Version of the US bicentennial emblem A pentagonal table by Bob Mackay [2] The Utah State Parks logo As impossible object ("Penrose" pentagram) Folded ribbon which is single-sided (more complex version of Möbius Strip). Non-pentagonal shape. Pentagram of circles. Alternate pentagram of intersecting circles. 3D-looking rendition. Partial view of US bicentennial logo on a shirt seen in Lisboa [3] Non-prime knot with two 5_1 configurations on a closed loop. Knotted epitrochoid Sum of two 5_1s, Vienna, orthodox church

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### Knot presentations

 Planar diagram presentation X1627 X3849 X5,10,6,1 X7283 X9,4,10,5 Gauss code -1, 4, -2, 5, -3, 1, -4, 2, -5, 3 Dowker-Thistlethwaite code 6 8 10 2 4 Conway Notation [5]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 5, width is 2,

Braid index is 2

[{7, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 1}]
 Knot 5_1. A graph, knot 5_1.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index 3 Nakanishi index 1 Maximal Thurston-Bennequin number [-10][3] Hyperbolic Volume Not hyperbolic A-Polynomial See Data:5 1/A-polynomial

### Four dimensional invariants

 Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $\textrm{ConcordanceGenus}(\textrm{Knot}(5,1))$ Rasmussen s-Invariant -4

### Polynomial invariants

 Alexander polynomial $t^2+ t^{-2} -t- t^{-1} +1$ Conway polynomial $z^4+3 z^2+1$ 2nd Alexander ideal (db, data sources) $\{1\}$ Determinant and Signature { 5, -4 } Jones polynomial $- q^{-7} + q^{-6} - q^{-5} + q^{-4} + q^{-2}$ HOMFLY-PT polynomial (db, data sources) $a^6 \left(-z^2\right)-2 a^6+a^4 z^4+4 a^4 z^2+3 a^4$ Kauffman polynomial (db, data sources) $a^9 z+a^8 z^2+a^7 z^3-a^7 z+a^6 z^4-3 a^6 z^2+2 a^6+a^5 z^3-2 a^5 z+a^4 z^4-4 a^4 z^2+3 a^4$ The A2 invariant $-q^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2 q^{10}+q^8+q^6$ The G2 invariant $q^{120}-q^{100}-q^{98}-q^{92}-q^{90}-q^{88}-q^{82}-q^{80}-q^{78}-q^{72}+q^{58}+q^{56}+q^{52}+2 q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+2 q^{40}+q^{38}+q^{34}+q^{32}+q^{30}$